Duos of Digital and Tangible Artefacts in Didactical Situations

Abstract

The duo of artefacts is a simplified model of the complex systems of various manipulatives (either tangible or virtual) that mathematics teachers and their students use in classrooms. It offers a means to study the complexity of the interweaving of the tangible and of the digital worlds in\xa0the teaching and learning processes. A duo of artefacts is defined as a specific combination of complementarities, redundancies and antagonisms between a tangible artefact and a digital artefact in a didactical situation. It is designed to provoke a joint instrumental genesis regarding both artefacts, and to control some of the schemes and mathematical conceptualizations developed by pupils during its use. This article exemplifies the model of a duo of artefacts, in the case of the pascaline and the e·pascaline for the learning of place-value base 10 notation of numbers. It details the design process of the e·pascaline (given the pascaline and its complementarities, redundancies and antagonisms), resulting from feedback of the digital environment and haptic properties of the tangible one. It provides examples of the evolution of pupils’ conceptions of numbers when using the duo. It also shows how teachers transform the duo into a system of instruments, allowing them to manage the problem-solving strategies of their students, providing them with one or the other artefact, playing with their complementarities and antagonisms.